find implicit differentiation (1-2xy^3)^5=x+4y

5 (1-2xy^3)^4 [-6xy^2dy-2y^3dx]=dx +4dy

[1+10
dx[(1-2xy^3)^4 y^3] =-dy[4+30(1-2xy^3)^4 xy^2]

dy/dx = -(1-2xy^3)^4y^3]/ [4+30(1-2xy^3)^4 xy^2]

check carefully

find the 4th derivative of 1/2x+3

To find the implicit differentiation of the equation (1 - 2xy^3)^5 = x + 4y, you need to differentiate both sides of the equation with respect to x. Implicit differentiation is used when the equation is not given explicitly as y = f(x).

Let's break down the process step by step:

Step 1: Start by differentiating both sides of the equation with respect to x.
d/dx((1 - 2xy^3)^5) = d/dx(x + 4y)

Step 2: Apply the chain rule to the left side of the equation by multiplying by the derivative of the inner function.
5(1 - 2xy^3)^4 * d/dx(1 - 2xy^3) = 1 + d/dx(4y)

Step 3: Simplify the equation by evaluating the derivatives.
5(1 - 2xy^3)^4 * (-2y^3 - 6x * y^2 * dy/dx) = 1 + 4 * dy/dx

Step 4: Rearrange the equation to solve for dy/dx, which represents the derivative of y with respect to x.
5(1 - 2xy^3)^4 * (-2y^3 - 6xy^2 * dy/dx) - 4dy/dx = 1

Step 5: Group the terms involving dy/dx on one side of the equation.
-10y^3(1 - 2xy^3)^4 - 24xy^2(1 - 2xy^3)^4 * dy/dx - 4dy/dx = 1

Step 6: Combine like terms.
-10y^3(1 - 2xy^3)^4 - (24xy^2(1 - 2xy^3)^4 + 4)dy/dx = 1

Step 7: Finally, solve for dy/dx.
dy/dx = (1 - 10y^3(1 - 2xy^3)^4) / (24xy^2(1 - 2xy^3)^4 + 4)

Now you have the result of implicit differentiation for the given equation, (1 - 2xy^3)^5 = x + 4y.